If, instead one specifies -a#b-, this term has (A-1) + (B-1) +
(A-1)*(B-1) , and is no longer partitioned into main effect of a,
main effect of b, and interaction. The omnibus test of this effect
is the overall test of the null hypothesis that there is
simultaneously no main effect of a, no main effect of b, and no a
by b interaction. As I show below, it simply tests the equality of
means in all of the cells. I think this is rarely of research
interest when one has this kind of "factorial" layout.

I agree, although there is probably somebody who likes this
parameterization. It does save you doing a little bit of algebra if
you want to get the mean for each cell in the
cross-classification. Incidentally, for me at least, the
equivalencies between the models are more easily seen by using the
coefficients than the odds ratios.

The other thing I don't like about this approach is that it confounds
the interaction with the main effects. If you instead do a##b, the
interaction term a*b gets presented separately, and you may see that
it is insignificant and not needed for the model, i.e. you only need
i.a and i.b and not a#b.

Clever people can probably figure out how to use lincom commands to
switch from one parameterization to another, but (unless it takes
forever for the model to run) it is probably easier just to run the
model different ways and let Stata do all the work.